## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Mary', 'hours worked by John', and 'hours worked by Hank', which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the natural language description into a symbolic representation
The symbolic representation of the variables is:
- $x_0$ : hours worked by Mary
- $x_1$ : hours worked by John
- $x_2$ : hours worked by Hank

## Step 3: Define the objective function in symbolic notation
The objective function to minimize is $9.69x_0 + 4.91x_1 + 9.83x_2$.

## 4: List the constraints in symbolic notation
The constraints are:
- $3x_0 \geq 3$ (Mary's paperwork competence rating is 3, but this seems to be an equality, so $3x_0 = 3$ or simply $x_0 \geq 1$ if considering a minimum requirement)
- $1x_0 \geq 1$ (Mary's organization score is 1, similarly, $x_0 \geq 1$)
- $7x_1 \geq 7$ (John's paperwork competence rating is 7, so $x_1 \geq 1$)
- $4x_1 \geq 4$ (John's organization score is 4, so $x_1 \geq 1$)
- $6x_2 \geq 6$ (Hank's paperwork competence rating is 6, so $x_2 \geq 1$)
- $8x_2 \geq 8$ (Hank's organization score is 8, so $x_2 \geq 1$)
- $3x_0 + 6x_2 \geq 36$
- $3x_0 + 7x_1 + 6x_2 \geq 36$
- $4x_1 + 8x_2 \geq 18$
- $1x_0 + 8x_2 \geq 16$
- $1x_0 + 4x_1 + 8x_2 \geq 36$
- $-5x_0 + 6x_2 \geq 0$
- $2x_1 - 8x_2 \geq 0$
- $3x_0 + 7x_1 \leq 110$
- $3x_0 + 7x_1 + 6x_2 \leq 63$

However, upon reevaluation for accuracy in representation:
- The correct interpretation directly from the problem statement should reflect:
  - $3x_0 = 3$ implies $x_0 = 1$ (equality for Mary's rating)
  - $1x_0 = 1$ implies $x_0 = 1$ (equality for Mary's score)
  - $7x_1 = 7$ implies $x_1 = 1$ (equality for John's rating)
  - $4x_1 = 4$ implies $x_1 = 1$ (equality for John's score)
  - $6x_2 = 6$ implies $x_2 = 1$ (equality for Hank's rating)
  - $8x_2 = 8$ implies $x_2 = 1$ (equality for Hank's score)

And other constraints as listed.

## 5: Provide the symbolic representation in the required format
```json
{
'sym_variables': [('x0', 'hours worked by Mary'), ('x1', 'hours worked by John'), ('x2', 'hours worked by Hank')],
'objective_function': '9.69*x0 + 4.91*x1 + 9.83*x2',
'constraints': [
    'x0 = 1',
    'x1 = 1',
    'x2 = 1',
    '3*x0 + 6*x2 >= 36',
    '3*x0 + 7*x1 + 6*x2 >= 36',
    '4*x1 + 8*x2 >= 18',
    'x0 + 8*x2 >= 16',
    'x0 + 4*x1 + 8*x2 >= 36',
    '-5*x0 + 6*x2 >= 0',
    '2*x1 - 8*x2 >= 0',
    '3*x0 + 7*x1 <= 110',
    '3*x0 + 7*x1 + 6*x2 <= 63'
]
}
```

## 6: Implement the optimization problem using Gurobi
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()
    
    # Define variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Mary
    x1 = model.addVar(name="x1", lb=0)  # hours worked by John
    x2 = model.addVar(name="x2", lb=0)  # hours worked by Hank

    # Set variable bounds based on initial conditions
    x0.start = 1
    x1.start = 1
    x2.start = 1

    # Objective function
    model.setObjective(9.69 * x0 + 4.91 * x1 + 9.83 * x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(x0 == 1, name="Mary_rating")
    model.addConstr(x1 == 1, name="John_rating")
    model.addConstr(x2 == 1, name="Hank_rating")
    model.addConstr(3 * x0 + 6 * x2 >= 36, name="min_paperwork")
    model.addConstr(3 * x0 + 7 * x1 + 6 * x2 >= 36, name="min_combined_paperwork")
    model.addConstr(4 * x1 + 8 * x2 >= 18, name="min_organization_combined")
    model.addConstr(x0 + 8 * x2 >= 16, name="min_organization")
    model.addConstr(x0 + 4 * x1 + 8 * x2 >= 36, name="total_organization")
    model.addConstr(-5 * x0 + 6 * x2 >= 0, name="balance_constraint1")
    model.addConstr(2 * x1 - 8 * x2 >= 0, name="balance_constraint2")
    model.addConstr(3 * x0 + 7 * x1 <= 110, name="max_paperwork_combined")
    model.addConstr(3 * x0 + 7 * x1 + 6 * x2 <= 63, name="max_total_paperwork")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("Hours worked by Mary: ", x0.varValue)
        print("Hours worked by John: ", x1.varValue)
        print("Hours worked by Hank: ", x2.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```